Mechanical Properties of Solids

Edge of the aluminium cube, L = 10 cm = 0.1 m, Area A = 0.01 $m^2$

Mass attached, m = 100 kg = 100 $*$ 9.8 = 980 N = Applied force F

Shear modulus $\eta$ = 25 GPa = 25 $*{10}^{9}Pa$

Shear modulus $\eta$ = Shear stress / Shear strain = $\frac{\frac{F}{A}}{\frac{?L}{L}}$ ,  $?L=\frac{F*L}{\eta *A}$ = $\frac{980*0.1}{25\mathrm{}*{10}^9*0.01}$ = 3.92 $*{10}^{-7}m$

Diameter of wires, d = 0.25 cm, radius, r = 0.125 cm

Cross-sectional area, $A_1$ = $A_2$ = $\pi *r^2=0.049{cm}^2$ = 4.908 $*{10}^{-6}$ $m^2$

Length of the steel wire, $L_1=1.5\mathrm{}\mathrm{m}$ , length of the brass wire, $L_2=1.0\mathrm{}\mathrm{m}$

Change in length of the steel wire $=?L_1,$ Change in length of the copper wire $=?L_2$

Total force exerted on the steel wire, $F_1$ = ( 4+6) kg = 10 kg = 98 N

Young's modulus of steel , $Y_1$ = $\frac{\frac{F_1}{A_1}}{\frac{?L_1}{L_1}}$ = 2.0 $*{10}^{11}$ Pa

$?L_1$ = $\frac{F_1}{A_1}*\frac{L_1}{Y_1}=\mathrm{}\frac{98*1.5}{4.908*{10}^{-6}*2.0*{10}^{11}}$ = 1.497 $*{10}^{-4}$ m

Similarly for brass wire, $F_2$ = 6 kg = 58.8 N, $Y_2$ = $0.91*{10}^{11}\mathrm{}\mathrm{P}\mathrm{a}$

$?L_2$ = $\frac{F_2}{A_2}*\frac{L_2}{Y_2}=\mathrm{}\frac{58.8*1.0}{4.908*{10}^{-6}*0.91*{10}^{11}}$ = 1.316 $*{10}^{-4}$ m

For a given stress, the strain in rubber is more than it is in steel, hence the Young's modulus of rubber is lesser than in steel. So the statement is False.

Shear modulus is the ratio of the applied stress to the change in the shape of a body. The stretching of a coil changes its shape. Hence, shear modulus of elasticity is involved in this process.

 = 2.2 $*{10}^{-4}{m}^3$

Material A has greater Young's modulus.

Material A is the strongest as it can withstand more strain than material B without fracture.

From the given graph, for the value stress 150 $*{10}^6$ N/ $m^2$ , the strain is 0.002

Young's modulus = $\frac{150\mathrm{}*{10}^6}{0.002}$ = 7.5 $*{10}^{10}$ N/ $m^2$

Yield strength is the maxium strength the material can withstand in elastic limit. From the graph, the yield strength is 300 $*{10}^6$ or 3 $*{10}^8\mathrm{N}/m^2$

Length of the steel wire, $L_1$ = 4.7 m

Area of cross-section of the steel wire, $A_1$ = 3.0 $*{10}^{-5}$ m2

Length of the copper wire, $L_2$ = 3.5 m

Area of cross-section of the copper wire, $A_2$ = 4.0 $*{10}^{-5}$ m2

Change in length, $\Delta {L=}_{}{L}_1-L_2$

Let the force applied = F

Young's modulus in steel wire,

$Y_1$ = $\frac{F_1}{A_1}*\frac{L_1}{\mathrm{\Delta }{L}_{}}$ ….(1)

Young's modulus in copper wire,

$Y_2$ = $\frac{F_2}{A_2}*\frac{L_2}{\mathrm{\Delta }{L}_{}}$ …….(2)

The ratio of Young's modulus

$\frac{\mathrm{}{Y}_1}{Y_2}$ = $\frac{F_1}{A_1}*\frac{L_1}{\mathrm{\Delta }{L}_{}}*\frac{A_2}{F_2}*\frac{\mathrm{\Delta }{L}_{}}{{\mathrm{L}}_2}$ = $\frac{L_1}{A_1}*\frac{A_2}{L_2}$ = $\frac{4.7*4*{10}^{-5}}{3*{10}^{-5}*3.5}$ = $\frac{18.8}{10.5}=1.79$